Novel waves structures for the nonclassical Sobolev-type equation in unipolar semiconductor with its stability analysis

In this study, the Sobolev-type equation is considered analytically to investigate the solitary wave solutions. The Sobolev-type equations are found in a broad range of fields, such as ecology, fluid dynamics, soil mechanics, and thermodynamics. There are two novel techniques used to explore the solitary wave structures namely as; generalized Riccati equation mapping and modified auxiliary equation (MAE) methods. The different types of abundant families of solutions in the form of dark soliton, bright soliton, solitary wave solutions, mixed singular soliton, mixed dark-bright soliton, periodic wave, and mixed periodic solutions. The linearized stability of the model has been investigated. Solitons behave differently in different circumstances, and their behaviour can be better understood by building unique physical problems with particular boundary conditions (BCs) and starting conditions (ICs) based on accurate soliton solutions. So, the choice of unique physical problems from various solutions is also carried out. The 3D, line graphs and corresponding contours are drawn with the help of the Mathematica software that explains the physical behavior of the state variable. This information can help the researchers in their understanding of the physical conditions.

and combined non-degenerate Jacobi elliptic wave function-like solutions 18 .Lu et al. considered the unstable nonlinear Schrödinger equation and used the modified form of the simple equation methods to gain solutions of the various types, namely trigonometric, rational, hyperbolic, and exponential functions 19 .Arshad et al. discussed the higher ordered nonlinear schrödinger equations analytically with the NMEDA approach and obtained various trigonometric solutions 20 .Inc et al. worked on the solitary wave solution of the Sawada-Kotera equation with two different techniques and obtained numerous solutions 21 .
We are considering the nonclassical Sobolev equation such as here denotes the real value of the function of the spatial variable x and t > 0 , µ is a real value except zero and q > 1 is a natural number.It explains the quasi-stationary approaches that occur in a unipolar semiconductor when the free charge current of source is supply.Here △= ∂ 2 ∂x 2 + ∂ 2 ∂y 2 denotes the higher dimensional spatial variable, q stands for the nonlinearity of the unknown function and we take q = 4 is used for this study.So, we can modify the nonclassical Sobolev Eq. (1) as given below, When we talk about analytical solutions in mathematics, we usually mean solutions that are stated in terms of mathematical functions, such as infinite series or other precisely defined mathematical expressions.Both series and explicit (closed-form) representations are possible in analytical solutions.Usually, these answers are obtained by applying different mathematical operations to the differential equations.Solutions that satisfy a particular differential equation exactly, devoid of any approximation, are referred to as exact solutions.These solutions could be explicit formulas or analytical solutions, which are expressed in terms of mathematical functions or series.Particularly when it comes to partial differential equations, precise solutions are frequently interchangeable with analytical solutions.
Alquran, M., et al., Heart-cusp and bell-shaped-cusp optical solitons for the complex Hirota model 22 , and multiplicative of dual-waves dual-mode Schrödinger with nonlinearity Kerr laws 23 .Jaradat, et al., discussed the numerical solutions of weak-dissipative two-mode perturbed Burgers' and Ostrovsky models 24 and solitary two-wave solutions for a new two-mode version of the Zakharov-Kuznetsov equation 25 .He also constructed the variety of physical structures to the generalized equal-width equation derived from Wazwaz-Benjamin-Bona-Mahony model 26 and the combination of Dark-Bright Binary-Soliton derived from the (2 + 1)-dimensional Nizhnik-Novikov-Veselov (TMNNV) equation 27 .
In this study we used two techniques namely as New Auxiliary Equation (NAE) technique and the Generalised Riccati Equation (GRE) mapping method.Depending on your preferences and the particular problem you are attempting to address, you can choose between the New Auxiliary Equation (NAE) technique and the Generalised Riccati Equation (GRE) mapping method.Both approaches are instruments for solving specific kinds of differential equations, and the situation at hand will determine how successful they are.Nonlinear ordinary differential equations (ODEs) can be solved by the GRE mapping method, which converts them into Riccati differential equations.When used properly, this approach can be very effective.It is especially helpful for various applications related to control theory, optimum control issues, and specific kinds of mathematical modelling.In certain situations, the differential equations may become simpler as a result of the GRE approach, making them easier to analyse and solve numerically.This method have different types of 27 abundant solutions.The NAE approach is useful in many engineering and physics contexts because it works effectively in situations when the differential equation's coefficients are constants.Second-order linear differential equations, such as those describing oscillatory and harmonic systems, can be solved well using this method.It might not be appropriate for handling time-varying coefficients or more complicated nonlinear differential equations.Moreover it have not verity of the solutions it contains only five types of solutions.
The Generalized Riccati Equation Mapping Method and the New Auxiliary Equation Mapping Method are both mathematical techniques used to find exact solitary wave solutions to various nonlinear partial differential equations (PDEs).Each method has its strengths and weaknesses.These methods can be applied to a wide range of nonlinear PDEs, making it versatile in solving various physical and mathematical problems.It provides a general framework for finding solitary wave solutions, which can be customized and adapted for specific PDEs.These methods can involve complex algebraic manipulations and may require specialized mathematical skills to apply effectively, especially for more intricate PDEs.Success in finding exact solutions depends on the specific PDE and its characteristics.There is no guarantee that it will work for all PDEs.The main innovations of the manuscript are that the non-classical Sobolev equation is under consideration analytically.The analytical solutions are carried out with two techniques.The stability of the model is also discussed.New families of the

Extraction of exact solutions
In this part, the specific solutions of Eq. (1) must be found, applying the transformation by converting PDE into ODE for = w(η) , here η = ζ 1 x + ζ 2 y + ζ 3 t .Where ζ 1 , ζ 2 , ζ 3 and w are the constants and a actual value function respectively.Hence, by replacing the above modification into Eq.(1), we receive the ODE develop as given below where w is a polynomial and ′ = d dη .Also, we take the solution of Eq. ( 2) and get the polynomials develop as 43,44 , where the constants 0 and j (i = 1,2,3,…M) that can be solve to be later,here ψ(η) is simplify the Reccati Eq. as given below.
Homogenous balancing principle can be applied to find the value of K in the previous Eq.( 3) and we can enter M = 1 in Eq. ( 4) Determine the derivatives of Eq. ( 6) by applying the Eq. ( 5) and replace in the Eq.(3).After simplifying, collect each coefficients of the identical power of ψ and set them then all to zero to gain a equations of system.Apply mathematica to deal with the system of calculation and gain the family of solution as, (3)   In the next section we find the solution by the help of modified auxiliary equation method.

Modified auxiliary equation technique
We take the solution of Eq. ( 3) and get the polynomials form as follows 45 , where the constants 0 , i and ν i (i = 1,2,3,…M) that can be solved after that, here ω(ζ ) is simplify the solution that is given below.
here, b 1 , b 2 , b 3 and u with u > 0 u = 1 are arbitrary constant that are determine later.Homogenous balancing principle can be applied to find the value of M in the previous Eq.( 3) and we can enter M = 1 in Eq. ( 34) Determine the derivatives of Eq. ( 36) by applying the Eq. ( 35) and replace in the Eq.(3).After simplifying, collecting the coefficients of the same power of ω (uζ )j and ω −(uζ )j and set them then equal to zero in all poly- nomials to gain a system of equations.Apply mathematica to deal with the system of calculation and gain the family of solution as,

Stability
This part examined at the sobolev type equations stability analysis.We determined the transformation as, Apply this value, we gain S is the constant and steady state solution of Eq. (1), now, Linearising in the form ε , we get Assume the following solution to the previous equation, where ϒ 1 , ϒ 2 and are the normalized wave number and frequency.Replace this transformation in Eq. ( 45), after solving the previous equation we gain the value of as, Since the value of the is imaginary, there will be an exponential growth in perturbation and no apparent decay in the superposition of the solution.This indicates an unstable dispersion.

Physical representation
Here, examined is the graphical behavior of solutions given different parameter choices.Various families of solutions including singular periodic, periodic, singular wave, shock wave, shock-singular, periodic-singular, complex solitary-shock, and double singular have been efficiently gained.The gained results are very helpful in understanding the nonlinear dynamics for the comparison of experimental and numerical solutions.The 3D, line graphs and corresponding contours are drawn with the help of the Mathematica software that explains the physical behavior of the state variable.The The singular soliton behavior is drawn in the Fig. 10.These soliton behaviors are very fruitful for the current (39) passing in the semiconductors that how the current travels from one place to another place.Diagrams of the initial and boundary values have been presented by Figs. 13, 14, 15, 16.The physical explanation of our findings may be useful as a tool for future research into nonlinear wave problems in applied science.

Selection of unique physical problem
In this section, we select the unique physical problems from the above solutions that help the researchers to take such problems for the sake of approximate solutions.Unique characteristics like as stability, non-dispersiveness, and the capacity to hold their form while travelling at constant speeds make exact soliton solutions to nonlinear partial differential equations intriguing solutions.In domains like nonlinear optics, plasma physics, and fluid  www.nature.com/scientificreports/dynamics, they are frequently employed to explain a variety of physical phenomena.Solitons behave differently in different circumstances, and their behaviour can be better understood by building unique physical problems with particular boundary conditions (BCs) and starting conditions (ICs) based on accurate soliton solutions.

Conclusions
In this study, the Sobolev-type equation is considered analytically to explored the exact solitary wave solutions.These types of equation have their own importance in applied sciences due to the involvement of the mixed third order derivative.The Sobolev-type equations are found in a broad range of fields, such as ecology, fluid dynamics, soil mechanics, and thermodynamics.To, obtained the explicit solitary wave solutions we apply the two novel techniques namely as; generalized Riccati equation mapping and modified auxiliary equation (MAE) methods.The different types of abundant families of solutions in the form of dark soliton, bright soliton, solitary wave solutions, mixed singular soliton, mixed dark-bright soliton, periodic wave, and mixed periodic solutions.Moreover, we also discussed the linear stability analysis for the underlying model.Also, the specific physical problems with specific boundary conditions (BCs) and initial conditions (ICs) are constructed that are based on exact soliton solutions that can be help us better understand how solitons behave in various situations.Thus, selecting distinct physical problems (ICs and BCS) are also constructed from a range of solutions.The unique physical problems are selected from numerious solutions that will help the researchers to check the nonlinear dynamics in an accurate way.The 3D, line graphs and corresponding contours are drawn with the help of the Mathematica software that explains the physical behavior of the state variable.

Type 1 :
When b 2 2 − 4b 3 b 1 > 0 and b 3 b 1 = 0 , then the twelve type of hyperbolic solutions exist such as,

Figure 13 .
Figure 13.Diagrams of the initial and boundary values w 2 (x, t).

Figure 14 .
Figure 14.Diagrams of the initial and boundary values w 1 4(x, t).

Figure 15 .
Figure 15.Diagrams of the initial and boundary values w 25 (x, t).

Figure 16 .Example 4 :
Figure 16.Diagrams of the initial and boundary values w 32 x, y, t .